The “alternative” representation we are going to apply is the spinor representation of \(\mathrm {SU}(2)\)  [1]}, developed from the \(\mathrm {U}(N)\)  formulation of LQG [2]}, [3]}, [4]}.
The quantum states under this representation are called the coherent spin network states (see below) [5]}.
After a concise review of its general construction, we will specialize in the coherent spin network state of a tetrahedron graph, whose evaluation on identity after changing the weight gives the SGF [6]}, [7]}.
We will construct a new Ponzano-Regge model with spinor variables, where the SGF serves as the vertex amplitude. Similar to the original one, it can also be seen as built with local amplitudes associated to the elementary bubbles.
